Polynomials - Prasolov

Algorithms and Computation
in Mathematics • Volume 11
Editors
Manuel Bronstein Arjeh M. Cohen
Henri Cohen David Eisenbud
Bernd Sturmfels



Victor V. Prasolov

Polynomials
Translated from the Russian by Dimitry Leites

123


Victor V. Prasolov
Independent University of Moscow
Department Mathematics
Bolshoy Vlasievskij per.11
119002 Moscow, Russia
e-mail:

Dimitry Leites (Translator)
Stockholm University
Department of Mathematics
106 91 Stockholm, Sweden
e-mail:

Originally published by MCCME
Moscow Center for Continuous Math. Education
in 2001 (Second Edition)

Mathematics Subject Classification (2000): 12-XX, 12E05

Library of Congress Control Number: 2009935697

ISSN 1431-1550
ISBN 978-3-540-40714-0 (hardcover)
ISBN 978-3-642-03979-9 (softcover)
DOI 10.1007/978-3-642-03980-5

e-ISBN 978-3-642-03980-5

This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication
of this publication or parts thereof is permitted only under the provisions of the German Copyright
Law of September 9, 1965, in its current version, and permission for use must always be obtained
from Springer. Violations are liable for prosecution under the German Copyright Law.
Springer is a part of Springer Science+Business Media
springeronline.com
© Springer-Verlag Berlin Heidelberg 2004, First softcover printing 2010
Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Typeset by the translator.
Edited and reformatted by LE-TeX, Leipzig, using a Springer LATEX macro package.
Cover design: deblik, Berlin

Printed on acid-free paper


Preface

The theory of polynomials constitutes an essential part of university courses
of algebra and calculus. Nevertheless, there are very few books entirely devoted to this theory.1 Though, after the first Russian edition of this book was
printed, there appeared several books2 devoted to particular aspects of the
polynomial theory, they have almost no intersection with this book.
1

2

The following classical references (not translated into Russian and therefore not
mentioned in the Russian editions of this book) are rare exceptions:
Barbeau E. J., Polynomials. Corrected reprint of the 1989 original. Problem
Books in Mathematics. Springer-Verlag, New York, 1995. xxii+455 pp.;
Borwein P., Erd´elyi T., Polynomials and polynomial inequalities. Graduate
Texts in Mathematics, 161. Springer-Verlag, New York, 1995. x+480 pp.;
Obreschkoff N., Verteilung und Berechnung der Nullstellen reeller Polynome.
(German) VEB Deutscher Verlag der Wissenschaften, Berlin 1963. viii+298 pp.
For example, some recent ones: Macdonald I. G., Affine Hecke algebras and orthogonal polynomials. Cambridge Tracts in Mathematics, 157. Cambridge University
Press, Cambridge, 2003. x+175 pp.;
Phillips G. M., Interpolation and approximation by polynomials. CMS Books in
Mathematics/Ouvrages de Math´ematiques de la SMC, 14. Springer-Verlag, New
York, 2003. xiv+312 pp.;
Mason J. C., Handscomb D. C. Chebyshev polynomials. Chapman & Hall/CRC,
Boca Raton, FL, 2003. xiv+341 pp.;
Rahman Q. I., Schmeisser G., Analytic theory of polynomials, London Math.
Soc. Monographs (N.S.) 26, 2002;

Sheil-Small T., Complex polynomials, Cambridge studies in adv. math. 75,
2002;
Lomont J. S., Brillhart J., Elliptic polynomials. Chapman & Hall/CRC, Boca
Raton, FL, 2001. xxiv+289 pp.;
Krall A. M., Hilbert space, boundary value problems and orthogonal polynomials. Operator Theory: Advances and Applications, 133. Birkh¨
auser Verlag, Basel,
2002. xiv+352 pp.;
Dunkl Ch. F., Xu Yuan, Orthogonal polynomials of several variables. Encyclopedia of Mathematics and its Applications, 81. Cambridge University Press,
Cambridge, 2001. xvi+390 pp. (Hereafter the translator’s footnotes.)


VI

Preface

This book contains an exposition of the main results in the theory of
polynomials, both classical and modern. Considerable attention is given to
Hilbert’s 17th problem on the representation of non-negative polynomials by
the sums of squares of rational functions and its generalizations. Galois theory
is discussed primarily from the point of view of the theory of polynomials, not
from that of the general theory of fields and their extensions. More precisely:
In Chapter 1 we discuss, mostly classical, theorems about the distribution
of the roots of a polynomial and of its derivative. It is also shown how to
determine the number of real roots to a real polynomial, and how to separate
them.
Chapter 2 deals with irreducibility criterions for polynomials with integer
coefficients, and with algorithms for factorization of such polynomials and for
polynomials with coefficients in the integers mod p.
In Chapter 3 we introduce and study some special classes of polynomials:
symmetric (polynomials which are invariant when the indeterminates are permuted), integer valued (polynomials which attain integer values at all integer

points), cyclotomic (polynomials with all primitive nth roots of unity as roots),
and some interesting classes introduced by Chebyshev, and by Bernoulli.
In Chapter 4 we collect a lot of scattered results on properties of polynomials. We discuss, e.g., how to construct polynomials with prescribed values
in certain points (interpolation), how to represent a polynomial as a sum
of powers of polynomials of degree one, and give a construction of numbers
which are not roots of any polynomial with rational coefficients (transcendental numbers).
Chapter 5 is devoted to the classical Galois theory. It is well known that
the roots of a polynomial equation of degree at most four in one variable can
be expressed in terms of radicals of arithmetic expressions of its coefficients.
A main application of Galois theory is that this is not possible in general for
equations of degree five or higher.
In Chapter 6 three classical Hilbert’s theorems are given: an ideal in a
polynomial ring has a finite basis (Hilbert’s basis theorem); if a polynomial f
vanishes on all common zeros of f1 , . . . , fr , then some power of f is a linear
combination (with polynomial coefficients) of f1 , . . . , fr (Hilbert’s Nullstellensatz); and if M = ⊕Mi is a finitely generated module over a polynomial ring
over K, then dimK Mi is a polynomial in i for large i (the Hilbert polynomial
of M ).
Furthermore, the theory of Gr¨obner bases is introduced. Gr¨
obner bases
are a tool for calculations in polynomial rings. An application is that solving systems of polynomial equations in several variables with finitely many
solutions can be reduced to solving polynomial equations in one variable.
In the final Chapter 7 considerable attention is given to Hilbert’s 17th
problem on the representation of non-negative polynomials as the sum of
squares of rational functions, and to its generalizations. The Lenstra-LenstraLov`
asz algorithm for factorization of polynomials with integer coefficients is
discussed in an appendix.


Preface

VII

Two important results of the theory of polynomials whose exposition requires quite a lot of space did not enter the book: how to solve fifth degree
equations by means of theta functions, and the classification of commuting
polynomials. These results are expounded in detail in two recently published
books in which I directly participated: [Pr3] and [Pr4].
During the work on this book I received financial support from the Russian
Fund of Basic Research under Project No. 01-01-00660.
Acknowledgement. Together with the translator, I am thankful to Dr.
Eastham for meticulous and friendly editing of the English and mathematics,
to J. Borcea, R. Fr¨oberg, B. Shapiro and V. Kostov for useful comments.
V. Prasolov
Moscow, May 1999


VIII

Preface

Notational conventions
As usual, Z denotes the set of all integers, N the subset of positive integers,
Fp = Z/pZ for p prime.
(Z/nZ)∗ denotes the set of invertible elements of Z/nZ.
|S| denotes the cardinality of the set S.
R[x] denotes the ring of polynomials in one indeterminate x with coefficients in a commutative ring R.
[x] denotes the integer part of a given real number x, i.e., the greatest
integer which is ≤ x.

Numbering of Theorems, Lemmas and Examples is usually continuous
throughout each section, e.g., reference to Lemma 2.3.2 means that the Lemma

is to be found in subsection 2.3 inside the same chapter 2.
Subsections are numbered separately, so Theorem 2.3.4 may occure in
subsec. 2.3.2.
Certain Lemmas and Examples (considered of local importance) are numbered simply Lemma 1, and so on, and, to find it, the page is indicated in the
reference.


Contents

1

Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Inequalities for roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . .
1.1.2 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Laguerre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Apolar polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 The Routh-Hurwitz problem . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The roots of a given polynomial and of its derivative . . . . . . . . .
1.2.1 The Gauss-Lucas theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 The roots of the derivative and the focal points of an
ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Localization of the roots of the derivative . . . . . . . . . . . . .
1.2.4 The Sendov-Ilieff conjecture . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 Polynomials whose roots coincide with the roots of
their derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 The resultant and the discriminant . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 The resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 The discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Computing certain resultants and discriminants . . . . . . .

1.4 Separation of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 The Fourier–Budan theorem . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Sturm’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Sylvester’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Separation of complex roots . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Lagrange’s series and estimates of the roots of a given
polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 The Lagrange-B¨
urmann series . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Lagrange’s series and estimation of roots . . . . . . . . . . . . .
1.6 Problems to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Solutions of selected problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
1
2
5
7
11
12
12
14
15
18
20
20
20
23
25

27
27
30
31
35
37
37
40
41
42


X

Contents

2

Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Main properties of irreducible polynomials . . . . . . . . . . . . . . . . . .
2.1.1 Factorization of polynomials into irreducible factors . . . .
2.1.2 Eisenstein’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Irreducibility modulo p . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Irreducibility criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Dumas’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Polynomials with a dominant coefficient . . . . . . . . . . . . . .
2.2.3 Irreducibility of polynomials attaining small values . . . .
2.3 Irreducibility of trinomials and fournomials . . . . . . . . . . . . . . . . .
2.3.1 Irreducibility of polynomials of the form xn ± xm ± xp ± 1
2.3.2 Irreducibility of certain trinomials . . . . . . . . . . . . . . . . . . .

2.4 Hilbert’s irreducibility theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Algorithms for factorization into irreducible factors . . . . . . . . . .
2.5.1 Berlekamp’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Factorization with the help of Hensel’s lemma . . . . . . . . .
2.6 Problems to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Solutions of selected problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
47
47
50
51
52
52
56
58
59
59
63
65
68
68
71
73
74

3

Polynomials of a Particular Form . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.1.1 Examples of symmetric polynomials . . . . . . . . . . . . . . . . . 77
3.1.2 Main theorem on symmetric polynomials . . . . . . . . . . . . . 79
3.1.3 Muirhead’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.4 The Schur functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Integer-valued polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.1 A basis in the space of integer-valued polynomials . . . . . 85
3.2.2 Integer-valued polynomials in several variables . . . . . . . . 87
3.2.3 The q-analogue of integer-valued polynomials . . . . . . . . . 88
3.3 The cyclotomic polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3.1 Main properties of the cyclotomic polynomials . . . . . . . . 89
3.3.2 The M¨
obius inversion formula . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.3 Irreducibility of cyclotomic polynomials . . . . . . . . . . . . . . 91
3.3.4 The expression for Φmn in terms of Φn . . . . . . . . . . . . . . . 93
3.3.5 The discriminant of a cyclotomic polynomial . . . . . . . . . . 94
3.3.6 The resultant of a pair of cyclotomic polynomials . . . . . . 95
3.3.7 Coefficients of the cyclotomic polynomials . . . . . . . . . . . . 96
3.3.8 Wedderburn’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3.9 Polynomials irreducible modulo p . . . . . . . . . . . . . . . . . . . . 99
3.4 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.1 Definition and main properties of Chebyshev
polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.2 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.3 Inequalities for Chebyshev polynomials . . . . . . . . . . . . . . . 107
3.4.4 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109


Contents

XI

3.5 Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.5.1 Definition of Bernoulli polynomials . . . . . . . . . . . . . . . . . . 112
3.5.2 Theorems of complement, addition of arguments and
multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.5.3 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.5.4 The Faulhaber-Jacobi theorem . . . . . . . . . . . . . . . . . . . . . . 117
3.5.5 Arithmetic properties of Bernoulli numbers and
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.6 Problems to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.6.1 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.6.2 Integer-valued polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.6.3 Chebyshev polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.7 Solution of selected problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4

Certain Properties of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1 Polynomials with prescribed values . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.1 Lagrange’s interpolation polynomial . . . . . . . . . . . . . . . . . 133
4.1.2 Hermite’s interpolation polynomial . . . . . . . . . . . . . . . . . . 136
4.1.3 The polynomial with prescribed values at the zeros of
its derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2 The height of a polynomial and other norms . . . . . . . . . . . . . . . . 139
4.2.1 Gauss’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.2.2 Polynomials in one variable . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.2.3 The maximum of the absolute value and S. Bernstein’s
inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.2.4 Polynomials in several variables . . . . . . . . . . . . . . . . . . . . . 148
4.2.5 An inequality for a pair of relatively prime polynomials 151
4.2.6 Mignotte’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.3 Equations for polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.3.1 Diophantine equations for polynomials . . . . . . . . . . . . . . . 154
4.3.2 Functional equations for polynomials . . . . . . . . . . . . . . . . . 162
4.4 Transformations of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.4.1 Tchirnhaus’s transformation . . . . . . . . . . . . . . . . . . . . . . . . 166
4.4.2 5th degree equation in Bring’s form . . . . . . . . . . . . . . . . . . 168
4.4.3 Representation of polynomials as sums of powers of
linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.5 Algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.5.1 Definition and main properties of algebraic numbers . . . 173
4.5.2 Kronecker’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.5.3 Liouville’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4.6 Problems to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179


XII

Contents

5

Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.1 Lagrange’s theorem and the Galois resolvent . . . . . . . . . . . . . . . . 181
5.1.1 Lagrange’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.1.2 The Galois resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.1.3 Theorem on a primitive element . . . . . . . . . . . . . . . . . . . . . 189
5.2 Basic Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2.1 The Galois correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2.2 A polynomial with the Galois group S5 . . . . . . . . . . . . . . 195
5.2.3 Simple radical extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

5.2.4 The cyclic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.3 How to solve equations by radicals . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.3.1 Solvable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.3.2 Equations with solvable Galois group . . . . . . . . . . . . . . . . 200
5.3.3 Equations solvable by radicals . . . . . . . . . . . . . . . . . . . . . . 201
5.3.4 Abelian equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.3.5 The Abel-Galois criterion for solvability of equations
of prime degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.4 Calculation of the Galois groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.4.1 The discriminant and the Galois group . . . . . . . . . . . . . . . 212
5.4.2 Resolvent polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.4.3 The Galois group modulo p . . . . . . . . . . . . . . . . . . . . . . . . . 216

6

Ideals in Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.1 Hilbert’s basis theorem and Hilbert’s theorem on zeros . . . . . . . 219
6.1.1 Hilbert’s basis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.1.2 Hilbert’s theorem on zeros . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.1.3 Hilbert’s polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.1.4 The homogeneous Hilbert’s Nullstellensatz for p-fields . . 231
6.2 Gr¨
obner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.2.1 Polynomials in one variable . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.2.2 Division of polynomials in several variables . . . . . . . . . . . 235
6.2.3 Definition of Gr¨
obner bases . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.2.4 Buchberger’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
6.2.5 A reduced Gr¨
obner basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7

Hilbert’s Seventeenth Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.1 The sums of squares: introduction . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.1.1 Several examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.1.2 Artin-Cassels-Pfister theorem . . . . . . . . . . . . . . . . . . . . . . . 248
7.1.3 The inequality between the arithmetic and geometric
means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.1.4 Hilbert’s theorem on non-negative polynomials p4 (x, y) . 253
7.2 Artin’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.2.1 Real fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.2.2 Sylvester’s theorem for real closed fields . . . . . . . . . . . . . . 263


Contents

XIII

7.2.3 Hilbert’s seventeenth problem . . . . . . . . . . . . . . . . . . . . . . . 266
7.3 Pfister’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
7.3.1 The multiplicative quadratic forms . . . . . . . . . . . . . . . . . . 270
7.3.2 Ci -fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
7.3.3 Pfister’s theorem on the sums of squares of rational
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
8.1 The Lenstra-Lenstra-Lov´asz algorithm . . . . . . . . . . . . . . . . . . . . . 279
8.1.1 The general description of the algorithm . . . . . . . . . . . . . 279

8.1.2 A reduced basis of the lattice . . . . . . . . . . . . . . . . . . . . . . . 280
8.1.3 The lattices and factorization of polynomials . . . . . . . . . . 283

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297



1
Roots of Polynomials

1.1 Inequalities for roots
1.1.1 The Fundamental Theorem of Algebra
In olden times, when algebraic theorems were scanty, the following statement
received the title of the Fundamental Theorem of Algebra:
“A given polynomial of degree n with complex coefficients has exactly n
roots (multiplicities counted).”
The first to formulate this statement was Alber de Girard in 1629, but he
did not even try to prove it. The first to realize the necessity of proving the
Fundamental Theorem of Algebra was d’Alembert. His proof (1746) was not,
however, considered convincing. Euler (1749), Faunsenet (1759) and Lagrange
(1771) offered their proofs but these proofs were not without blemishes, either.
The first to give a satisfactory proof of the Fundamental Theorem of Algebra was Gauss. He gave three different versions of the proof (1799, 1815
and 1816) and in 1845 he additionally published a refined version of his first
proof.
For a review of the different proofs of the Fundamental Theorem of Algebra, see [Ti]. We confine ourselves to one proof. This proof is based on the
following Rouch´e’s theorem, which is of interest by itself.
Theorem 1.1.1 (Rouch´
e). Let f and g be polynomials, and γ a closed curve
without self-intersections in the complex plane1 . If

f (z) − g(z) < f (z) + g(z)

(1)

for all z ∈ γ, then inside γ there is an equal number of roots of f and g
(multiplicities counted).
1

The plane C1 of complex variable.

V.V. Prasolov, Polynomials, Algorithms and Computation in Mathematics 11,
DOI 10.1007/978-3-642-03980-5_1, © Springer-Verlag Berlin Heidelberg 2010

1


2

1 Roots of Polynomials

Proof. In the complex plane, consider vector fields v(z) = f (z) and
w(z) = g(z). From (1) it follows that at no point of γ are the vectors v
and w directed opposite to each other. Recall that the index of the curve γ
with respect to a vector field v is the number of revolutions of the vector v(z)
as it completely circumscribes the curve γ. (For a more detailed acquaintance
with the properties of index we recommend Chapter 6 of [Pr2].) Consider the
vector field
vt = tv + (1 − t)w.
Then v0 = w and v1 = v. It is also clear that at every point z ∈ γ the vector
vt (z) is nonzero. This means that the index ind(t) of γ with respect to the

vector field vt is well defined. The integer ind(t) depends continuously on t,
and hence ind(t) = const. In particular, the indices of γ with respect to the
vector fields v and w coincide.
Let the index of the singular point z0 be defined as the index of the curve
|z − z0 | = ε, where ε is sufficiently small. It is not difficult to show that the
index of γ with respect to a vector field v is equal to the sum of indices of
singular points, i.e., those at which v(z) = 0. For the vector field v(z) = f (z),
the index of the singular point z0 is equal to the multiplicity of the root z0 of
f . Therefore the coincidence of the indices of γ with respect to vector fields
v(z) = f (z) and w(z) = g(z) implies that, inside γ, the number of roots of f
is equal to that of g. ✷
With the help of Rouch´e’s theorem it is not only possible to prove the
Fundamental Theorem of Algebra but also to estimate the absolute value of
any root of the polynomial in question.
Theorem 1.1.2. Let f (z) = z n + a1 z n−1 + · · · + an , where ai ∈ C. Then,
inside the circle |z| = 1 + max |ai |, there are exactly n roots of f (multiplicities
i

counted).
Proof. Let a = max |ai |. Inside the circle considered, the polynomial
i

g(z) = z n has root 0 of multiplicity n. Therefore it suffices to verify that,
if |z| = 1 + a, then f (z) − g(z) < f (z) + g(z) . We will prove even that
f (z) − g(z) < g(z) , i.e.,
|a1 z n−1 + · · · + an | < |z|n .
Clearly, if |z| = 1 + a, then
|a1 z n−1 + · · · + an | ≤ a |z|n−1 + · · · + 1 = a

|z|n − 1

= |z|n − 1 < |z|n .
|z| − 1

1.1.2 Cauchy’s theorem
Here we discuss Cauchy’s theorem on the roots of polynomials as well as its
corollaries and generalizations.


1.1 Inequalities for roots

3

Theorem 1.1.3 (Cauchy). Let f (x) = xn − b1 xn−1 − · · · − bn , where all
the numbers bi are non-negative and at least one of them is nonzero. The
polynomial f has a unique (simple) positive root p and the absolute values of
the other roots do not exceed p.
Proof. Set
bn
f (x)
b1
+ · · · + n − 1.
=
xn
x
x
If x = 0, the equation f (x) = 0 is equivalent to the equation F (x) = 0. As
x grows from 0 to +∞ the function F (x) strictly decreases from +∞ to −1.
Therefore, for x > 0, the function F vanishes at precisely one point, p. We
have
f (p)

b1
nbn
− n = F (p) = − 2 − · · · − n+1 < 0.
p
p
p
F (x) = −

Hence p is a simple rooe algorithm that computes a
reduced basis of the lattice. We therefore first discuss the notion of a reduced
basis of a lattice and the algorithm of calculating the reduced basis (this
algorithm was also suggested in [Le2], and therefore is also called an LLLalgorithm).
8.1.2 A reduced basis of the lattice
A subset L ⊂ Rn is called a lattice of rank n if there exists a basis b1 , . . . , bn
in Rn such that
n

L=
i=1

Z · bi =

n

ri bi | ri ∈ Z .

i=1

The determinant of the lattice L is the number
d(L) = det(b1 , . . . , bn ) ,

where bi denotes the column-vector of coordinates of the vector bi . The determinant of the lattice is equal to the volume of the parallelepiped spanned by
b1 , . . . , bn . There are several bases of the lattice but the determinant of transition from one basis to another is equal to ±1, and so the number d(L) > 0
does not depend on the choice of a basis.


8.1 The Lenstra-Lenstra-Lov´
asz algorithm

281

Let b1 , . . . , bn be a basis of L. The Gram–Schmidt orthogonalization yields
an orthogonal (not necessarily orthonormal) basis
i−1

b∗i = bi −

µij b∗j ,

where

µij =

j=1

(bi , b∗j )
, 1 ≤ j < i ≤ n.
(b∗j , b∗j )

A basis b1 , . . . , bn of the lattice L is called reduced if
|µij | ≤

1
3
for 1 ≤ j < i ≤ n and |b∗i + µi,i−1 b∗i−1 |2 ≥ |b∗i−1 |2 for 1 < i ≤ n.
2
4

Theorem 8.1.1 (Hadamard’s inequality). The volume of a given parallelepiped does not exceed the product of the length of its edges, i.e.,
n

|bi |.

d(L) ≤
i=1

Proof. The vectors b∗i are orthogonal, and so
i−1

|bi |2 = |b∗i |2 +

µ2ij |b∗j |2 ≤ |b∗i |2 .
j=1

n

Moreover, d(L) =
i=1

|b∗i |. ✷

In the following theorem we have collected the main inequalities for the
vectors of a reduced basis.
Theorem 8.1.2. Let b1 , . . . , bn be a reduced basis of the lattice L. Then:
a) d(L) ≤

n

|bi | ≤ 2n(n−1)/4 d(L)
i=1
2(i−1)/2 |b∗i | for 1 ≤ j ≤ i
(n−1)/4
1/n

.

b) |bj | ≤
≤ n.
c) |b1 | ≤ 2
d(L) .
d) If x ∈ L and x = 0, then |b1 | ≤ 2(n−1)/2 |x|.
e) If the vectors x1 , . . . , xt ∈ L are linearly independent, then
|bj | ≤ 2(n−1)/2 max |x1 |, . . . , |xt | for 1 ≤ j ≤ t.
Proof. a) From the definition of a reduced basis it follows that
|b∗i |2 ≥
since |µi,i−1 | ≤ 12 .

1
3
− µ2i,i−1 |b∗i−1 |2 ≥ |b∗i−1 |2 ,
4

2


282

Appendix

A simple induction shows that |b∗j |2 ≤ 2i−j |b∗i |2 for i ≥ j. Therefore
i−1

|bi |2 = |b∗i |2 +

µ2ij |b∗j |2 ≤ |b∗i |2 1 +
j=1

1+2+· · ·+2i−2
2

= |b∗i |2

1+2i−1
.
2

Hence
n

|bi |2 ≤
i=1

1+2n−1
1+1 1+2
·
·. . .·
2
2
2

n

|b∗i |2 ≤ 1·2·. . .·2n−1 d(L)2 = 2n(n−1) d(L)2 .
i=1

b) From the inequalities
|b∗j |2 ≤ 2i−j |b∗i |2 for i ≥ j
|bj |2 ≤

1 + 2j−1 ∗ 2
|bj | ≤ 2j−1 |b∗j |2
2

we deduce that
|bj |2 ≤ 2i−j · 2j−1 |b∗j |2 = 2i−1 |b∗i |2 for i ≥ j.
c) Thanks to b) we have
|b1 |2 ≤ |b∗1 |2 ,
|b1 |2 ≤ 2|b∗2 |2 ,
.................
|b1 |2 ≤ 2n−1 |b∗n |2 .
The product of these inequalities yields
n

|b∗i |2 = 2(n−1)/2 d(L)2 .

|b1 |2n ≤ 2n(n−1)/2
i=1
n

n

ri bi =

d) Express x in the form x =
i=1

i=1

si b∗i , where ri ∈ Z and si ∈ R.

Let i0 be the greatest index for which ri = 0. Then ri0 = si0 . Hence
|x|2 ≥ s2i0 |b∗i0 |2 = ri20 |b∗i0 |2 ≥ |b∗i0 |2 ≥ 21−i0 |b1 |2 ≥ 21−n |b1 |2 .
e) The vectors x1 , . . . , xt cannot all belong to the subspace spanned by
b1 , . . . , bt−1 . Hence, for a vector xs , we have |xs | ≥ |b∗i |2 , where i ≥ t (see d)).
Therefore, for j ≤ i, we obtain
|xs |2 ≥ |b∗i |2 ≥ 2j−i |b∗j |2 ≥ 2j−i · 21−j |bj |2 = 21−i |bj |2 ≥ 21−n |bj |2 .


8.1 The Lenstra-Lenstra-Lov´
asz algorithm

283

Let us now describe the algorithm for calculating a reduced basis of the
lattice L. Suppose that the vectors b1 , . . . , bk−1 form a reduced basis of the
lattice they span (we start with one vector, i.e., with k = 2). We adjoin
to them a vector bk which belongs to L but does not lie in the subspace
spanned by b1 , . . . , bk−1 . The construction of the basis of the lattice generated
by b1 , . . . , bk is performed as follows.
(bi ,b∗ )
Step 1(Fulfilling the condition |µkj | ≤ 12 .) Recall that µij = (b∗ ,bj∗ ) . Start
j

j

with l = k and suppose that |µkj | ≤ 12 for l < j < k. Replace bk by bk − qbl ,
where q is the integer nearest to µkl . This transformation preserves µkj for
j > l (since b∗j ⊥ bl for l < j) and replaces µkl by µkl −q since (bl , b∗l ) = (b∗l , b∗l ).
Clearly, |µkl − q| ≤ 12 , and so after such a modification of the bi the condition
|µkj | ≤ 12 will be satisfied for l − 1 < j < k. Now repeat the operation.
Step 2 (Fulfilling the condition |b∗k |2 ≥ ( 34 − µ2k,k−1 )|b∗k−1 |2 .) Suppose that
3
|b∗k |2 < ( − µ2k,k−1 )|b∗k−1 |2 .
4
Then we replace the ordered set (b1 , . . . , bk−2 , bk−1 , bk ) by the ordered set
(b1 , . . . , bk−2 , bk , bk−1 ). Under this change b∗k−1 will be replaced by b∗k +
µk,k−1 b∗k−1 , and so |b∗k−1 |2 will be replaced by
|b∗k |2 + µ2k,k−1 |b∗k−1 |2 <

3
3
− µ2k,k−1 |b∗k−1 |2 + µ2k,k−1 |b∗k−1 |2 = |b∗k−1 |2 .

4
4

Let us consider the reduced basis b1 , . . . , bk−2 and apply the first step of the
algorithm to it. Since |b∗k−1 |2 decreases it follows that the algorithm converges
(for the rigorous proof of the convergence of the algorithm, see [Le2] and
[Co3]).
8.1.3 The lattices and factorization of polynomials
We recall that it remains to construct an algorithm which, for an irreducible
divisor h (mod p) of the polynomial f (mod p) without multiple divisors,
computes an irreducible divisor h0 of f for which h0 (mod p) is divisible by
h (mod p). In doing so, we may assume that h is a monic polynomial.
In intermediate calculations we have to consider divisibility modulo pk .
We therefore consider a more general case: let us assume that h (mod pk ) is
an irreducible divisor of the polynomial f (mod pk ) without multiple divisors,
the polynomial h is monic and h0 is an irreducible divisor of f for which h0
(mod p) is divisible by h (mod p). It is easy to verify that in this case h0
f
(mod pk ) is divisible by h (mod pk ). Indeed, the polynomial
(mod p) is
h0
not divisible by the irreducible polynomial h (mod p), i.e., these polynomials
are relatively prime. Therefore
λh + µ

f
= 1 − pν
h0

for some λ, µ, ν ∈ Z[x].

284

Appendix

Let us multiply both sides of this equality by (1 + pν + · · · + pk−1 ν k−1 )h0 . As
a result, we obtain λ1 h + µ1 f ≡ h0 (mod pk ). But f (mod pk ) is divisible by
h (mod pk ), and so h0 (mod pk ) is divisible by h (mod pk ).
Let n = deg f and l = deg h. Fix an integer m ≥ l and consider the set of
all polynomials with integer coefficients of degree not greater than m which
are divisible by pk modulo h. As we have just shown, h0 belongs to this set if
deg h0 ≤ m.
To g(x) = a0 + · · · + am xm , we assign the point (a0 , . . . , am ) ∈ Rm+1 .
Under this map the polynomials considered form a lattice L. Clearly, the
a2i is just the Euclidean length in Rm+1 .
norm |g| =
The basis of the lattice L consists of the polynomials pk xi , where 0 ≤ i < l,
and the polynomials h(x)xj , where 0 ≤ j ≤ m − l. The coordinates of these
vectors (polynomials) relative to the basis 1, x, . . . , xm constitute a matrix
pk Il ∗
, where Il is the l × l unit matrix and I is an upper
of the form
0 I
triangular matrix with units on the main diagonal. Hence d(L) = pkl .
Before we start describing the algorithm for calculating the polynomial h0 ,
we prove two theorems which provide the estimates we need.
Theorem 8.1.3. If a polynomial b ∈ L is such that |b|n · |f |m < pkl , then b is
divisible by h0 . (In particular, (f, b) = 1, where (f, b) is the greatest common
divisor of f and b.)

Proof. We may assume that b = 0. Set g = (f, b). We would like to prove
that g is divisible by h0 . To this end, it suffices to prove that g (mod p) is
divisible by h (mod p). Indeed, if g (mod p) is divisible by h (mod p) and
f
f
∈ Z[x], then
is not divisible by h0 because h (mod p) is a simple (of
g
g
multiplicity 1) divisor of f (mod p). Hence g is divisible by h0 .
Suppose that g (mod p) is not divisible by h (mod p). The polynomial h
(mod p) is irreducible, and so g (mod p) and h (mod p) are relatively prime,
i.e., there exist polynomials λ1 , µ1 , ν1 ∈ Z[x] such that
λ1 h + µ1 g = 1 − pν1 .

(1)

Let e = deg g and m = deg b. Clearly, 0 ≤ e ≤ m ≤ m. Set
M = λf + µb | λ, µ ∈ Z[x], deg λ < m − e, deg µ < n − e
⊂ Z + Z · x + · · · + Z · xn+m −e−1 .
Denote by M the image of M under the natural projection onto
Z · xe + · · · + Z · xn+m −e−1 .
First we show that, if the image of λf + µb ∈ M is equal to 0 ∈ M , then
λ = µ = 0. Indeed, in this case deg(λf + µb) < e, but λf + µb is divisible by g


8.1 The Lenstra-Lenstra-Lov´
asz algorithm

and deg g = e. Hence λf + µb = 0, i.e., λ

f
g

= −µ

285

b
. The polynomials
g

b
f
f
and are relatively prime, and so µ is divisible by . But deg µ < n − e =
g
g
g
f
deg . Hence µ = 0, and therefore λ = 0.
g
Thus, the projections of the sets {xi f | 0 ≤ i < m − e} and {xj b | 0 ≤ j <
n − e} onto M are, on the one hand, linearly independent and, on the other
hand, generate M . This means that the projections of these two sets form a
basis of the lattice M . In particular, the rank of the lattice M is equal to
n + m − 2e. Moreover, by Hadamard’s inequality d(M ) ≤ |f |m −e |b|n−e .
By the hypothesis, |f |m |b|n < pkl . Since m ≤ m, we have
d(M ) ≤ |f |m |b|n < pkl .

(2)

Let us show that, if ν ∈ M and deg ν < e + l, then p−k ν ∈ Z[x]. Indeed, the
polynomial ν = λf + µb is divisible by g = (f, b), and so having multiplied
ν
(1) by (1 + pν1 + · · · + pk−1 ν1k−1 ) we obtain
g
ν
λ2 h + µ2 h ≡
(mod pk ).
(3)
g
We consider the situation when f (mod pk ) is divisible by h (mod pk ). Further, b ∈ L, and so b (mod pk ) is divisible by h (mod pk ). Since ν ∈ M , it
follows that ν = λf + µb, so ν (mod pk ) is divisible by h (mod pk ). Therefore
(3) implies that ν/g (mod pk ) is also divisible by h (mod pk ). But
deg

ν
g

(mod pk )


and h (mod pk ) is a monic polynomial of degree l. Hence

ν
≡ 0 (mod pk ),
g

and therefore ν ≡ 0 (mod pk ).
In M , we can select a basis be , be+1 , . . . , bn+m −e−1 such that deg bj = j.
It is easy to verify that e + l − 1 ≤ n + m − e − 1. Indeed, the polynomial b is
f
(mod p) is divisible by
divisible by g, and so e = deg g ≤ deg b = m ; also
g
h (mod p), and so
l = deg h ≤ deg f − deg g = n − e.
The elements be , . . . , be+l−1 ∈ M are obtained from the polynomials that lie in
M and are divisible by pk under the projection that annihilates terms of degree
lower than e. Therefore all the coefficients of the polynomials be , . . . , be+l−1
(in particular, their highest coefficients) are divisible by pk .
Clearly, the discriminant d(M ) is equal to the absolute value of the product of the highest coefficients of the polynomials be , . . . , bn+m −e−1 , and so it
is not less than the product of the absolute values of the highest coefficients
of be , . . . , be+l−1 . Hence d(M ) ≥ pkl . This contradicts inequality (2). ✷


286

Appendix

In the next theorem we, as earlier, assume that h is a monic polynomial
and f (mod pk ) is divisible by h (mod pk ); L is the lattice of polynomials
of degree not higher than m and divisible by h modulo pk ; l = deg h and
n = deg f ; h0 is an irreducible divisor of f for which h0 (mod p) is divisible
by h (mod p), i.e., h0 (mod pk ) is divisible by h (mod pk ).
Theorem 8.1.4. Let b1 , b2 , . . . , bm+1 be a reduced basis of the lattice L. Suppose that
n/2
2m

|f |m+n .
pkl > 2mn/2
m
a) Then deg h0 ≤ m if and only if
|b1 | <

pkl
.
|f |m

n

b) Suppose that, for a basis vector bj , we have
|bj | <

pkl
.
|f |m

n

(∗)

Let t be the greatest of all such indices j. Then
deg h0 = m + 1 − t;

h0 = GCD(b1 , . . . , bt );

and inequality (∗) holds for j = 1, . . . , t.
pkl

, i.e., |b1 |n · |f |m < pkl . Then
|f |m
by Theorem 8.1.3, the polynomial b1 ∈ L is divisible by h0 . On the other
hand, the condition b1 ∈ L implies that deg b1 ≤ m. Thus deg h0 ≤ m.
Now suppose that deg h0 ≤ m, i.e., h0 ∈ L. By Corollary of Mignotte’s
Proof. a) First, suppose that |b1 | <

theorem (see page 154), |h0 | ≤
x = h0 , we obtain

2m
m

n

|f |. By applying Theorem 8.1.2 (d) for

2m
|f |.
m

b1 ≤ 2m/2 |h0 | ≤ 2m/2
By the hypothesis, 2mn/2

2m n/2
|f |n
m

2m/2

<

pkl
|f |m ,

2m
|f | <
m

(4)

i.e.,

n

pkl
.
|f |m

Formulas (1) and (5) yield the inequality desired: |b1 | <

(5)

n

pkl
.
|f |m

8.1 The Lenstra-Lenstra-Lov´
asz algorithm

287

b) Let J be the set of all the indices j for which (∗) holds. By Theorem
8.1.3, if j ∈ J, then bj is divisible by h0 . Therefore h1 = GCD(bj | j ∈ J) is
divisible by h0 . Here if j ∈ J, then bj is divisible by h1 and deg bj ≤ m, i.e.,
bj belongs to the lattice
Z · h1 + Z · h1 x + · · · + Z · h1 xm−deg h1 .
Since the vectors b1 , . . . , bm are linearly independent, it follows that
|J| ≤ m + 1 − deg h1 ,

(6)

where |J| is the cardinality of J.
By Corollary of Mignotte’s theorem (see page 154), we have
2m
|f | for any i ≥ 0.
m

|h0 xi | = |h0 | ≤

By definition, i = 0, 1, . . . , m − deg h0 for h0 xi ∈ L and these vectors are
linearly independent. Theorem 8.1.2 (e) is therefore applicable to them:
|bj | ≤ 2m/2 |h0 xi | ≤ 2m/2

By the hypothesis, 2m/2

2m

m

2m
|f | for
m
|f | <

n

1 ≤ j ≤ m + 1 − deg h0 .

pkl
, and so
|f |m

{1, 2, . . . , m + 1 − deg h0 } ⊂ J.
Since h1 is divisible by h0 , it follows that deg h1 ≤ deg h0 , and therefore
|J| ≥ m + 1 − deg h0 ≥ m + 1 − deg h1 .

(7)

By comparing the inequalities (6) and (7) we see that deg h0 = deg h1 = t
and J = {1, 2, . . . , t}.
It remains to verify that h0 = ±h1 . Since h0 is a divisor of the polynomial
f with content 1, it follows that cont(h0 ) = 1. Let j ∈ J and dj = cont(bj ).
bj
is also
By Theorem 8.1.3, the polynomial bj is divisible by h0 . Therefore
dj
bj

divisible by h0 . Since h0 ∈ L, we deduce that
∈ L. But bj is a basis
dj
element of the lattice L, and so dj = 1. This means that cont(h1 ) = 1, since
bj is divisible by h1 . Thus h1 is divisible by h0 and cont(h1 ) = 1, and so
h0 = ±h1 . ✷
Now we are able to describe the algorithm for calculating the polynomial
h0 .